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12/26/09 8:00 PMHow to Solve the Rubik's Cube
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There are three translations of this page: Danish (Dansk) (Word Document), Japanese (日本語) (HTML) and Portuguese (Português) (HTML). If you want to translate this page, goahead. Send me an email when you are done and I will add your translation to this list.
So you have a Rubik's Cube, and you've played with it and stared at it and taken itapart...need I go on any further? The following are two complete, fool-proof solutions tosolving the cube from absolutely any legal position. Credit goes not to me, but to DavidSingmaster, who wrote a book in 1980, Notes on Rubik's Magic Cube, which explainspretty much all of what you need to know, plus more. Singmaster wrote about all of thesemoves except the move for Step 2, which I discovered independently (along with manyother people, no doubt).
I've updated this page to include a second solution to the cube. I learned this solutionfrom the Handbook of Cubik Math by Alexander H. Frey, Jr. and David Singmaster. Iwould strongly recommend getting this book; there are all sorts of interesting problemsand exercises to do. This solution is a bit more free-form than the first solution, as thereare less moves to memorize. However, it probably requires a little more intuition aboutcubing on your part to be able to use it effectively. I'm not really sure which of the twosolutions would be better to learn first. I prefer the second method, because it is definitelysimpler.
I should point out that these are both beginner-level solutions. They are easy to learn (inparticular Solution #2). If you want to be able to solve the cube in 20 seconds, this is notthe page for you: check out the speed cubing links later on in the page. The fastersolutions generally require considerably more memorization and practice. If this is yourfirst time solving the cube, I think you've come to the right place.
I have a couple of "cheat sheets" (Solution #1, Solution #2), which are one pagedocuments, suitable for printing, that contain the moves. These are not suitable forlearning with, rather they can be used as a quick reminder while you are trying tomemorize the moves. They should print on one page in most browsers at normal text
Eliason CorporationChoose from a variety of Energy-Saving products offered by Eliason.www.eliasoncorp.com
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Good Luck CharmsAmulets and charms for love,money, and more. Absolutelyguaranteed!www.calastrology.com
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settings. If they don't fit, just reduce the size of the text in the browser and try again.These are new as of December 2003 so I am soliciting feedback (rubiks [at] jeays [dot]net)
Table of ContentsI. Solution #1 (Standard Solution)
Step 0 -- NotationStep 1 -- Do the first faceStep 2 -- Do the middle layer edge piecesStep 3 -- Form cross on last layerStep 4 -- Rotate U face edge piecesStep 5 -- Position U face corner piecesStep 6 -- Rotate corner pieces
II. Solution #2 (Easier to Memorize Solution)Background for Solution #2Step 1 -- U face edge piecesStep 2 -- U face corner piecesStep 3 -- Middle edge piecesStep 4 -- Solve remaining edge piecesStep 5 -- Position corner piecesStep 6 -- Orient corner pieces correctly
III. Frequently Asked QuestionsHow do I disassemble / assemble my cube?Why must the cube be reassembled correctly?How many (legal) positions does the cube have?What are some records related to solving the Rubik's Cube?Where do I purchase a Rubik's Cube?I'm bored, what do I do now?What are some links to other Rubik's Cube pages?What are some games I can play with the cube?How do I orient the center pieces?How do I tell if the cube is unsolvable from a given state?How do I solve a Sudokube?
IV. MiscellaneousPersonal RecordsPage HistoryPoll ArchivesWeb Tools That Reference This PageContact Me
Solution #1Step 0 -- Notation
Before we get started, we must work out a method for describing the various moves thatwill be made. There are six faces, with the following notations:
Upper, or top face = U
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Down, or bottom face = DLeft face = LRight face = RFront face = FBack face = B
We can turn each face either clockwise or counter-clockwise, with respect to the center(i.e. a move that may be clockwise to you, when looking at the cube, may not beclockwise for that face, in relation to the middle of the cube). The names for the differentkind of moves (I'll use the U face as an example) are:
A 90-degree turn clockwise on a face is denoted by U.A 90-degree turn counter-clockwise on a face is denoted by U' ("U prime") (Alsonote this is the same as U, done three times).A 180-degree turn either clockwise or counter-clockwise on a face, is denoted by U2
("U squared") and is the same as two clockwise turns, or two counter-clockwiseturns.
We can refer to individual pieces by a two-letter (for edges) or three-letter (for corners)combination. For example, the piece in the upper right front corner is called URF, and theedge piece to the down and left of the cube is called DL. Also, these notations refer to thepiece that is in that place at that time, not the piece that should go there.
Clockwise and counter-clockwise are also used to describe orientations of corner pieces.For the URF piece, for example, rotating it clockwise would result in the U side of thepiece on the R face, the R side on the F face, and the F side on the U face. Similarly, for acounter-clockwise rotation, the U side of the piece would end up on the F face, the F sideon the R face and the R side on the U face.
Also, note that during any sequence of moves the position of the center pieces with respectto one another is unchanged.
Step 1 -- Do the first face
I was thinking of omitting this step; if you've come so far as to seek help for doing theRubik's cube, then you've probably been able to do one side. But I've included the detailsin for completeness.
The first step involves choosing a color, and getting all the pieces of that color to form acomplete face. These must also be in the correct relative location.
First, we will do the edges. Let's say we pick white as the side to complete first (manypeople choose white as it is the easiest color to pick out). Turn the cube so that the whitecenter piece is on the U face. Note that the centers are attached, so that they are always inthe same position with respect to one another, unlike a cube of even degree (i.e. 2 x 2 x 2or 4 x 4 x 4).
First, we put the edge pieces (those with 2 colors) in the right place. There are severalpossibilities. Note that when moving pieces around you should have the piece that you areworking on the F face.
White/Other color (OC) piece is on the D face. Rotate the D face so that it is
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directly underneath the place where it needs to go (on the U face). If the white sideof the piece is on the D face, apply F2, and it will be correct. If the white side of thepiece is not on the D face, apply D R F' R'.White/OC piece is on the center slice (i.e. middle portion of the cube) . Apply F orF' to get the piece on the D face, and then perform above moves.White/OC piece is in correct position, but incorrectly rotated. Apply F2 and do theabove moves as necessary.
Second, we do the corners. There are six main possibilities for each of the four cornerpieces:
Corner piece in correct place. Do nothing.Corner piece is in the correct place, but incorrectly rotated, so that the piece needsrotating clockwise. Hold the cube so that this piece is in the URF location, andapply F D F' D' F D F'.Corner piece is in the correct place, but incorrectly rotated, so that the piece needsrotating counter-clockwise. Hold the cube so that this piece is in the URF location,and apply R' D' R D R' D' R.Corner piece on D face. Rotate D face so that the corner piece you want to moveinto position is directly underneath its intended location. If the corner piece haswhite (or whatever color you chose) on the bottom, and the destination of the pieceis URF (i.e. the upper right hand corner on the front of the cube), apply R' D2 R DR' D' R.Corner piece on D face, and white side of the corner is on the left hand side,assuming that the piece is going to the URF location. Apply F D F'.Corner piece on D face, and white side of the corner is on the right hand side,assuming that the piece is going to the URF location. Apply R' D' R.
You should be finished the white face by now. This step is fairly intuitive, so it shouldn'thave been too much of a problem.
Step 2 -- Do the middle layer edge pieces
Okay, so the U face is done...Now we have to get the edge pieces of the middle layer inplace, that is the FR, FL, BR, BL pieces.
There are four possibilities:
Edge piece is in the correct place and oriented correctly -- do nothing.Edge piece is in correct place, but oriented incorrectly. In this case, you have toreplace it with another piece temporarily. Choose an edge piece that is on the Dface, and move it to the position of the aforementioned edge piece using the movedescribed in the following list item.Edge piece is in an incorrect place in the middle layer. In most cases, you can justskip this edge piece for now, as it will move to the D face when you put the correctedge piece in that place. If you have two edge pieces in the middle layer that are ineach other's correct places, then you will need to break the cycle by replacing one ofthose edge pieces with one of the D face edge pieces.Edge piece is on the D face. This is good. Let's say you want to move the yellow-red edge piece into place on the middle layer. (Your cube may not have yellow andred as adjacent pieces, so you may have to substitute different colors). Check which
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side of the yellow-red piece is actually on the D face. Say yellow is on the D face.Rotate the D face so that the yellow-red piece is opposite from the yellow center. (Ifthe yellow center is on the F face, then the yellow-red piece is the DB piece, got it?). Now, hold the cube so that white is the U face, and the yellow center is on the Fface. Which side of the yellow-red piece is the red face on? If the red center is onthe L face, then apply F' D' F D L D L'. If the red center is on the R face, then applyF D F' D' R' D' R. One way to think about this move, if you're trying to memorize it,is that you are removing the corner piece on the U side right above the edge pieceyou're trying to replace, and then putting it back in from the other direction, with theside effect of moving the edge piece into the correct place.
This will have to be repeated at least 4 times in order to get all of the 4 middle layer edgepieces into place.
Step 3 -- Form cross on last layer
The first two layers should now be completely correct. From now on, turn the cube upside-down, so that the first face (white, in my example) is the D face. It will remain this wayuntil the cube is complete. The reason for this is just to make the manipulations a biteasier to perform.
On my cube, green is opposite to white, so I will refer to the green face as being the newU face for our purposes. We must now try to form a green cross, out of the green center,and the four edge pieces that surround it.
There are 4 possibilities:
All four pieces are correct, and you have green cross on the U face. Do nothing.Two adjacent edge pieces are correct. For example, the UF and UR piece havegreen on the top, forming a sort of L shape, with the center. The UL and UB piecesdo not have green on the top. In this situation, hold the cube so that the UF and URpieces have green on the top (as in the example earlier in this paragraph), and applyB U L U' L' B'.Two opposite edge pieces are correct, and the other two aren't. Let's say UR and ULare correct (this should make a green line down the middle of the U face). Apply BL U L' U' B'.No edge pieces have green on the top. Hold the cube any way (still keeping thegreen center on the top) and apply B L U L' U' B' F U R U' R' F' (for those of youwho want to memorize the moves, you should realize that this is the same as doingthe move in the third part of this section, then rotating the cube 180 degrees, andthen applying the move in the second part of this section).
You should now have the bottom two layers all correct, as before, and a green cross (orwhatever the color on your cube whose center is opposite to white is) on the top face.
IMPORTANT: If the bottom two faces are perfectly correct, and you have 1 or 3edge pieces that show green on the top, then your cube is messed up. Somebody(maybe you!) took it apart (or changed the stickers around) at some point in time.Might as well take it all apart right now, reassemble it correctly and start again.
Step 4 -- Rotate U face edge pieces
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You now have a green cross on the U face, but...these edge pieces may not be in thecorrect order. This step involves rotating them so that the can be lined up with theirrespective colors.
If all 4 pieces are correct (the entire cube is correct except for four corner pieces onthe top layer), then do nothing.If 1 piece is correct, then rotate the whole cube so that this piece is in the ULposition (make sure green is still on the top). If the remaining 3 edge pieces need tobe rotated clockwise, apply R2 D' U2 R' L F2 R L' D R2. If the pieces need to berotated counter-clockwise, apply R2 D' R' L F2 R L' U2 D R2.If 2 "adjacent" edge pieces (by "adjacent" here I mean pairs such as UF and UL, orUB and UL, and not UF and UB, or UR and UL), then rotate the U face so that only1 edge piece is correct, and follow that rule.If 2 opposite (i.e. not adjacent!) edge pieces are correct, then apply U or U' andfollow the rule for 0 edge pieces correct.If 0 edge pieces are correct, turn the U face so that: for the UF piece (which can beany piece), the F side of that piece is the same color as the R face center. Nowapply R2 D2 B2 D L2 F2 L2 F2 L2 F2 D' B2 D2 R2.
The four U face edge pieces should now be in the correct place. The cube should be allcorrect now except for the four corner pieces on the U face.
Step 5 -- Position U face corner pieces
Now we must move the corner pieces into the right places.
There are 3 possibilities:
All 4 corner pieces are in the right place, although not necessarily rotated correctly.Do nothing.1 corner piece is in the right place. Hold the cube so that it is in the UFR position. Ifthe remaining three corner pieces need to be rotated clockwise, apply L' U R U' L UR' U'. If they need to be rotated counter-clockwise, apply U R U' L' U R' U' L.0 corner pieces are in the right place. If they all need to go to opposite corners,apply R' B2 F R F' R' F R F' R' F R F' R' B2 R. If they need to go to adjacentcorners, hold the cube so that UFR and UFL pieces need swapping, and so do theUBR and UBL pieces. Apply B L U L' U' L U L' U' L U L' U' B'.
Okay, now all of the corner pieces should be in the right place. We're almost there!
Step 6 -- Rotate corner pieces
Hold the cube so that an incorrectly rotated corner piece is in the UFR position. If it needsto be rotated clockwise, apply F D F' D' F D F' D'. If it needs to be rotated counter-clockwise, apply D F D' F' D F D' F'. Now (and this is extremely important) turn the Uface only, so that the next incorrectly rotated corner piece is in the UFR position. Applyone of the above moves, depending on which way it needs rotating. Repeat if more thantwo corner pieces are incorrectly rotated. After all pieces are rotated, simply turn the Uface and complete the cube.
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This step may be confusing, simply because after just one corner piece is rotated the cubeis in quite serious disarray. Just make certain that you move only the U face and it shouldwork out fine. I'll give one example to show exactly what will happen, for a simpleexample: the UFR piece needs rotating clockwise, and the UBR piece needs rotatingcounter-clockwise. Do these moves and the cube will be complete F D F' D' F D F' D' U DF D' F' D F D' F' U'.
Note that there are only certain combinations of incorrectly rotated pieces. If your cubegets to a position where there is a situation other than one of these, then it has probablybeen taken apart by small green aliens, so I would recommend disassembling it andstarting over.
One piece needs rotating clockwise, one piece needs rotating counter-clockwise,other two are correct.Three pieces all need rotating clockwise, other one is correct.Three pieces all need rotating counter-clockwise, other one is correct.Two pieces need rotating clockwise, two pieces need rotating counter-clockwise.I was going to write "all corner pieces are correct" here, but I guess that would beobvious.
Solution #2Background for Solution #2
If you are not familiar with the standard notation, please check Step 0 -- Notation fromSolution #1. The following solution is a much easier to memorize way of solving the cube.Most of the moves are easy to understand, such that you don't even feel that you arememorizing anything. I will try to explain what to think when you are solving the cube.
This is new as of December 2006. A lot of people have asked for pictures or videosexplaining the moves, so I've joined the crowd at YouTube with some tutorial videoson cubing. I hope they are useful. Check out my How to Solve the Rubik's CubePlaylist on YouTube. Note that this explains Solution #2 only.
Step 1 -- U face edge pieces
So your cube is scrambled right now. The first thing to do is to chose a color, say white (ittends to stand out from the other colors on the cube). It's also a very good idea to alwaysto a specific color first, since you will begin to learn which colors are adjacent, whichspeeds up things considerably.
The first step is to form a cross on the top face of the cube. Orient the cube so that thewhite center piece is on top. You want to get the correct pieces in the UL, UB, UR andUF locations. So, you will need to done some of the following moves: (be sure to do thosein the first step first.
If a white-other color (OC) piece is on the U face:If white is in the U position, simply rotate the U face until the OC is lined upwith its center.
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If OC is in the U position, rotate the U face so that the piece is at an adjacentedge location to its desired location. Hold your cube so that white is the Ucenter and OC is the F center. Now rotate U so that the white-OC piece is inthe UR position. Now apply R' F'.
If a white-OC piece is in the middle slice of the cube (the middle third), then holdthe cube so that white is still on the U face, but this white-OC piece is in the FRlocation. Now, you should notice that you will be able to move it to the U face byapplying F' (if the white face on the R side) or R (if the white face is on the F side).Find the spot where that white-OC piece should go. Rotate U until you can applyeither F' or R to move the white-OC piece in the correct spot, so that the white facewill move to the top. Examples: You want to move the piece in FR, with whitebeing the R face, to its home location at UL, so apply U' F' U. You want to moveFR, with white being the F face, to its home location at UL, so apply U2 R U2. Seehow you simply move U, then bring the edge piece up to the U face, then move Uback to restore the original position, plus the piece you just moved.If a white-OC piece is on the bottom slice of the cube:
If the white is on the D face, simply rotate D until the OC is directlyunderneath its center, and apply F2 (assuming the piece is at the FD position)to put it in the correct location.If the OC is on the D face, hold the cube so white is the U center, and OC isthe F center. Rotate D so that the white-OC piece is in the RD position, andapply R F' R' (you do not need R' if the UR piece had not been placedcorrectly yet).
You should now have a white cross formed on the top of your cube. Also you should bedeveloping an intuition about these moves. What you will learn to do after a few timesthrough this, is just think how the edge pieces are located relative to one another. Thisshould speed things up.
Step 2 -- U face corner pieces
The second step is to correctly position three of the U face corner pieces. The reason thatyou will only put three of them and not four into place is that this method uses a "workingspace" which greatly simplifies the later steps.
There are three basic possibilities for putting corner pieces into place:
The piece is on the D slice, with the white side not on the D side. In this case, rotatethe D face so that it is directly underneath the location that it should go to. Now,hold the cube so that the piece is in the DRF spot, and the intended location is theUFR spot.
If white is on the R side of the corner piece in DRF, apply R' D' R.If white is on the F side of the corner piece in DRF, apply F D F'.
The corner piece is on the D slice, but the white face is on the D side. Rotate the Dface so that the corner piece is in the DRF spot, and the intended location is theURF spot. Now apply R' D2 R D R' D' R. Note that you are doing R' D2 R to movethe white side off the bottom of the cube, so that you can use one of the moves inthe previous section. Also note that equivalent to this is: F D2 F' D' F D F'. I wouldimagine if you are left-handed this would make things easier. Speaking of which, ifyou are left-handed, I would be interested if you naturally use a particular one of
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these processes. Personally I am right-handed and do the "R' D2 R..." move, withoutreally thinking about it.The corner piece in question is in the right spot but incorrectly rotated. Therefore,we must rotate it. Hold the cube so that it is in the URF location. Now,
If the white side is on the R face, apply R' D' R D R' D' R.If the white side is on the F face, apply F D F' D' F D F'.
So now you should be done one side, except for one corner piece. This location will beused to swap corner pieces in and out, greatly simplifying later processes. The moves inthe first two steps are really quite intuitive. After only a few repetitions, you should findthem simple and natural to do.
Step 3 -- Middle edge pieces
This step involves correctly placing three of the four edge pieces on the "middle" layer ofthe cube. For these moves you will need to hold your cube so that the white face is on thebottom. The only middle layer edge piece that you do not position is the one right abovethe corner piece that you did not position correctly in step 2.
First of all, make sure the white side is on the bottom, and the "empty" (i.e. incorrect)corner piece on the white side is in the DRF location. The middle layer edge pieces will allbe positioned in this step, except for the FR one.
To move a piece into position, rotate the cube about its vertical axis, so that the intendedlocation is the FR location. (For example, you want to put the FL piece in place. Rotatethe cube a quarter turn counter-clockwise). Now rotate the bottom slice so that theincorrect corner piece is in the DRF location. (So in the previous example -- for the FLpiece -- you would first turn the cube, then apply D').
Now you are ready to do the move. The move to put the new edge piece into place canonly be done if it is on the U slice. If it is, note which side is NOT on the U face. You willneed to apply either F' or R, depending on the orientation of the edge piece you want tomove. Then, apply U until the piece you want to move is in the UF or UR (depending onyour previous move) location, and then F or R', to get it back to normal. I'd better give anexample... Yellow is the F center. Orange is the R center. You wish to position theYellow-Orange edge piece, to the FR position. You have already rotated the D face so thatthe DRF location does not contain a white corner piece. You see the Yellow-Orange piecein the UB location. You note that Orange is the U side, and Yellow is the B side. Thus,you apply F' U2 F. All that description for three easy moves :-).
To continue, simply keep rotating D or D' and moving the cube to set up the sameposition, with an "empty" corner in DRF, the intended location at RF, and the piece youwant to move in the U slice. Note that in some cases the piece may already be in thecorrect location, but orientated incorrectly. In this case you will have to take it out first(i.e. put any edge piece with the color whose center is opposite white on your cube intothat location) and then put it back in that spot. In other words, with the DRF corner"empty" and the offending piece in the FR spot, apply F' U' F U R U' R'.
Now you should be done 2/3 of the cube, less two pieces: a middle layer edge piece and itsadjacent corner piece, that appears to take a chunk out of the bottom (white) layer. Notethat it is possible for the "empty" corner piece on the bottom layer to get solved by
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accident. If so, just ignore it, and pretend that it is unsolved.
Step 4 -- Solve remaining edge pieces
This is the only step that requires any actual memorization. I think you'll find that themoves from the other steps become very natural after a short time. There are two basicparts to this step, as follows. The goal of the whole step is to solve all of the 5 remainingedge pieces. The first part is to solve three of these (UF, UL, UB), and the second part isto solve the other two together.
First of all, hold the cube so that the "empty" edge piece is in the BR position, and thusthe "empty" corner piece is in the RDB position. To do moves in this part, you first of allmove a piece into the BR location, then move it to the U face, to one of those UF, UL, orUB positions. The move is as follows. First, optionally rotate U so that either the UR orUB location is lined up next to the BR location correctly. Then, apply R' or B such thatafter you do the move, the U face color of that BR edge piece moves to the U face. Thenrotate U so that the edge piece you were moving is in the correct location. Then do R or B'(to undo the first part of this move). Pretty simple huh?! An example may be in order.Let's say the Blue-Yellow piece is in the BR location. Furthermore, Blue is the U color,and Yellow is the L color. You would thus apply U [to put the UL location (thedestination) in the right spot] B U' B'. However, when actually trying to solve the cubequickly, before applying U' in the previous move, you should look to find the next edgepiece that you wish to put in the right location. So rotate U until it is in the UB location,and then by applying B' you return the cube to a stable position. Then, you will need torotate U some amount to get the UL piece (Blue-Yellow in the example) back to the rightplace. There is a tremendous amount of freedom in this sequence of moves. In fact, you donot need to return the edge pieces to the correct spots in between repetitions of this move.Simply recognize how the pieces go with respect to one another, and then finally alignthem, when all three (UF, UL, UB) are done.
Also, note that sometimes the BR piece is not one of the U face pieces. In this case, youwill have to put it where the edge piece that has the U and R face colors is, and bring theremaining U face edge piece that you need to solve into the BR location.
Now, there are four possibilities. The remaining edge pieces are the BR piece and the URpiece. Do the following:
Luckily, the pieces are correct. Move to the next step and smile at your goodfortune.The pieces are in the correct locations, but incorrectly oriented. Apply B U' B' U R'U R U'.Both edge pieces (BR and UR) have the same color on the R side of the piece,which is the same color as the R center. Apply U' R' U' R U' R' U' R U'.The other case (the UR piece has the R color on its U side, and B color on its Rside, and the BR piece has the U color on its R side, and the R color on its B side).Apply B U B' U B U B' U2.
If you want to reduce memorization at the expense of some speed, two of these movessuffice. In other words, if you apply all three of these moves in any sequence to an all-edges correct cube, you will get back an all-edges correct cube.
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Step 5 -- Position corner pieces
In this step you want to move the remaining unsolved corner pieces to their correctlocations, irrespective of orientation. Hold the cube so that the "empty" corner piece on thebottom (white) face is in the DRB location. Rotate the U face so that the piece that youwant to position is in the UFL location. Apply L D2 L'. Now, rotate the U face so that thelocation (with respect to the top-layer edge pieces) of the corner piece you are working onis in the UFL location. Now apply L D2 L' again. Rotate U so that everything lines up.Repeat this (up to 3 times) until all of the corner pieces are in the right location.
This step is a little confusing at first. First of all, make sure your DRB piece is that"empty" (unsolved...not missing :-) ) corner piece. Say the UFL piece is Blue-Yellow-Orange. But that piece should go in the URB location. You would do the following moves:L D2 L' [move the piece in question out of the way (to the DRB location, if you'reinterested)] U' [move the correct location to the UFL spot] L D2 L' [move the piecequestion back to the U slice] U [undo the U twist you did earlier]. One thing to note whendoing this move, make sure the original UFL piece does not contain the color of thebottom face (white in my ongoing example). Also note that you are free to rotate the Uface before the move so that you can move a particular corner piece that you want toposition into the UFL location so that you can work with it. The only (slight) differencewill be that you will need to rotate U at the end to make up for that. Note that these U-rotations should be very obvious. You can simply line up the top-layer edge pieces withtheir respective centers.
One other thing to note for this step is that after finishing this step, the DBR piece maystill require rotating. This is different from Solution #1 if you are familiar with that. Thiswill be handled in Step 6.
The remaining paragraphs in this section are optional, and a bit more advanced andlengthy, so you may want to skip over them if this is your first time through. If you wantto add a bit of speed to this solution and perhaps save some work during Step 6, read on.There are two ways of optimizing this step. First, you can position two of the cornerpieces at once (instead of one, as I describe above), and second, you can often rotate someof the pieces appropriately while you are positioning them.
If you consider a plane of symmetry running through the four corners (yes, only three areactually needed) UFL, UBR, DFL, DBR, you will notice that L D2 L' (the move Iintroduced at the start of this section) and F' D2 F are mirror images. This means it'spossible to do either one (the same move must be done twice, you can't mix and match) inorder to complete this section. If you know how the pieces are going to rotate, then youcan take advantage of this in order to do some orienting of the pieces while you positionthem.
I usually approach this section by first looking at the DBR piece. This allows you toposition two pieces at once, since you don't just choose an arbitrary corner on the U face,you specifically rotate U so that when you start the move, you will send the DBR piece tothe proper location. There are three possibilities for how it is rotated.
If the U color is on the R face of the DBR piece, then doing the F' D2 F move willcorrectly rotate it. So in this case, you would first of all position the U face so thatthe UFL position was where the DBR piece should go, then do F' D2 F, then rotate
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U so to piece you just moved out of the UFL position will get moved into thecorrect place, and then do F' D2 F again.If the U color is on the B face of the DBR piece, then doing the L D2 L' move willcorrectly rotate it. Follow the same procedure as for the previous condition.If the U color is on the D face of the DBR piece, then you can't correctly rotate itwith one of the moves. However, you can still correctly position two corner pieces,if the place that you are sending the DBR piece to is not occupied by a corner piececontaining white, the D color.
If the DRB piece is already in the correct place, you can often save a rotation by choosingto correctly position a U face corner piece that has the U color on its U face. Both the F'D2 F move or L D2 L' move will rotate it correctly.
Step 6 -- Orient corner pieces correctly
Corner pieces must be rotated in pairs -- one clockwise and one counter-clockwise. Findtwo incorrectly rotated corner pieces that are on the same slice. Hold the cube so that oneof the pieces in the UFL position and the other is somewhere on the U slice.
To rotate a piece clockwise, apply L D2 L' F' D2 F.To rotate a piece counter-clockwise, apply F' D2 F L D2 L'.
After orienting the first corner piece, apply U until the other corner piece moves to theUFL location. You will then need to turn U to undo the previous twisting (this should befairly obvious). Here's an explicit example -- the UFL piece needs rotating counter-clockwise, and the UFR piece needs rotating clockwise. The full sequence would be asfollows: F' D2 F L D2 L' [orient UFL piece] U [position other corner] L D2 L' F' D2 F[orient original UFR piece] U' [undoes rotation of U that was done earlier].
You may need to apply this pattern up to 3 times, however if you use the strategy that Iexplain in Step 5 of choosing between the two equivalent moves, you will normally needto do do this move 0 to 2 times. Note that you can only do one clockwise and one counter-clockwise twist. You cannot twist three corner pieces all clockwise, like the corner-twisting move in Solution #1. You always have to do them in pairs. Here's an example,let's say the UFL, UFR and UBR pieces all need rotating clockwise. The full sequence torotate all three corners would be as follows: L D2 L' F' D2 F [orient UFL corner] U [moveoriginal UFR piece to working UFL corner] F' D2 F L D2 L' [turn original UFR counter-clockwise] F' D2 F L D2 L' [turn original UFR counter-clockwise again, solving it] U[move original UBR piece to working UFL corner] L D2 L' F' D2 F [orient original UBRcorner] U2 [realign U face].
If the two remaining corner pieces are diametrically opposed (e.g. at UFL and DRB), thenyou can apply R2 (in this case) to bring both of them onto the U slice. Then, do thesequence. The apply R2 again to get to the original configuration.
Frequently Asked Questions
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How do I disassemble / assemble my cube?
So you want/need to take your cube apart. Turn a face 45 degrees. Obtain a fairly flat keyor screwdriver. Gently lift the edge piece in the middle of the rotated face with yourthumb, while gently inserting the screwdriver. Slowly prise the piece out. Do not force it.After one piece is out the rest come out fairly easily. You might want to take a good lookat the mechanism that holds the cube together; it's quite interesting. If you were unclearbefore about how the centers always have the same relative position, it should be veryclear now.
Time to put the cube back together? Assemble the first two layers correctly. This isn't toobad. Then for the top layer, first turn the center piece 45 degrees, then insert one edgepiece, then the two surrounding corner pieces, then the adjacent edge pieces, so that onerow is left without pieces. Then place the two corner pieces in their positions. For the finaledge piece, hold it at about a 45 degree angle, and push it gently down into place.
Why must the cube be reassembled correctly?
Note that you must reassemble the cube so that it is solved correctly. If you reassemble thepieces haphazardly, there is but a 1 in 12 chance that your cube will be solvable.Wondering why? The following is not a rigorous proof but rather an explanation of whatis happening. There are 12 edge pieces, and eight corner pieces in a cube. These moveabout separately, that is to say, an edge piece could never swap places with a corner piece.It's possible to move all but one of the edge pieces to any location, with either orientation.The last edge piece is forced to take on a particular orientation. You can see this foryourself by simply removing one edge piece and turning it around. If you now try to solvethe cube, you will find that it is impossible. However, if you remove another (any other!)edge piece and flip this one too, and try to solve this cube, you will find that it is possible.Thus, there are two equally likely possibilities for the edge pieces when the cube isassembled randomly: edge pieces have been flipped from their true positions an evennumber of times, in which case the cube is solvable; or an odd number of times, in whichcase it's not.
Onto the corner pieces. Each corner piece has three possible orientations: correct, rotatedclockwise, and rotated counter-clockwise. Of the 8 corner pieces, they must have a totalnumber of rotations that is divisible by three evenly. You can test this for yourself bytwisting one corner piece, and trying to solve the cube. You'll find that it's impossible. Ifyou twist another random piece the opposite way, these will cancel each other out and thecube will be solvable, however if it gets twisted the same way, the cube will still beunsolvable. You'll need a third twist in the same direction to return the cube to a solvableposition. So, if all the corner pieces are in place, there are three equally likely situations tooccur. Finally, we must look at the actual positions of the corner pieces, without regard torotation. Notice how during step 5, it is never the case that only two corner pieces are outof position? Well, that is one of the unsolvable positions. The "first" six corner pieces cango anywhere. This leaves two corner pieces, for two corner spots. One of these positions issolvable, and one of them isn't.
The three events above are all independent, so applying a little probability, we can see thatthere are 2 x 3 x 2 = 12 "orbits" or groups of positions that cannot be reached from oneanother. Only one of these leads to the solved Rubik's Cube.
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How many (legal) positions does the cube have?
The naive way to approach this problem is as follows: there are 12 edge pieces, and 8corner pieces. First of all, note that (obviously) a corner piece could never go in the spotof an edge piece. Edge pieces and corner pieces can be arranged in 12! * 8! ways,according to basic counting laws. Now, each of the edge pieces can have one of twoorientations, and each of the corner pieces can have one of three orientations. So we mustmultiply the previous number (counting the different locations of pieces), by 212 * 38,representing the total number of positions. This number is 519,024,039,293,878,272,000(~= 5.19 * 1020). However, this is not the correct answer, due to physical constraints ofthe cube. Not all of these positions are possible, as outlined in the previous section. Weare able to rotate all but one of the corner pieces in any way, but the final corner piece isdetermined by the first seven. So we must divide our final answer by 3 (i.e. the number oforientations we originally gave to the final corner piece) to account for this. The sameargument holds for the edge pieces, so we must also divide by 2. Finally, we cannot swaptwo edge pieces or two corner pieces. If you decide to take apart your cube, and simplyswap two edge pieces, you will find the cube unsolvable. What will happen is that you cansolve the edge pieces, but then two corner pieces will be swapped, and this situationcannot be fixed. So, we divide the total number again by 2 (i.e. we divide the abovenumber by 2 * 3 * 2 = 12 -- "from" the previous section). This result, the "order" of thecube "group", is: 43,252,003,274,489,856,000 (~= 4.32 * 1019). I like to give a physicalrepresentation to big numbers, so here goes: if you had a cube for every legal position,then you could cover the entire surface of the earth (including oceans :-) ) about 250times. A chain consisting of all the cube positions would stretch about 250(coincidentally!) light years. Feel free to invent your own comparisons. Also, "order" and"group" have specific mathematical definitions which, alas, I still don't understand.
What are some records related to solving the Rubik's Cube?
The current official record for solving a Rubik's Cube is 10.48 seconds, achieved in 2006by Toby Mao. In unofficial competition, or in practice, there are a number of people whohave recorded faster times than these (see the Speed Cubing links below).
The original, oft-quoted world record that has been in the Guinness Book of WorldRecords, is 22.95 seconds, set by Minh Thai, of Vietnam, in the original Rubik's CubeWorld Championship held in 1982.
Check RecordHolders.org for official records and Speedcubing.com for many unofficialrecords related to solving the Rubik's Cube.
Where do I purchase a Rubik's Cube?
When I originally wrote this page in 1995, finding a Rubik's Cube wasoften something of a challenge. However since then many reliablesources for Rubik's Cubes have sprung up. If you live in NorthAmerica, your best bet is probably to visit the nearest Toys "R" Us.They normally have Rubik's Cubes in stock. I strongly recommendgetting an official Rubik's Cube (branded with rubiks.com). There are
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some imitation cubes floating around and they tend to be of poorquality.
I bought an imitation Rubik's Cube from the "It Store" at the RideauCenter in Ottawa, Ontario, Canada (my hometown). The mechanism isa little stiff and the colors are glittery and non-standard (silveropposite gold, pink opposite red and blue opposite green). Themanufacturer is Funworks and the Hologram Rubik's Cube cost isC$7.99. I wouldn't particularly recommend this brand but it would doin a pinch. As of August 2002 their site seems to have severalannoying Javascript errors so it may be hard to find what you want.Also in Ottawa, Mrs. Tiggy Winkles has Rubik's products in stock.
I have been informed that Dollarama in Toronto sells imitation Rubik'sCubes for C$1.00. It's unlikely that the quality is very good but theprice is certainly right. Another Toronto location with Rubik's Cubesis Toys Toys Toys in the Eaton Center. The cost is C$19.99 and itcomes with a book. I have also been told that Cracker Barrel in theUSA sells imitation Rubik's Cubes (called "Magic Cubes") for $4.99US. The mechanism is a little stiff but solid.
There's a link to Amazon.com on the side here to an official Rubik'sCube. Full disclosure: I get a kickback of approximately 42 cents ifyou click on the link and buy through the site (there is no extra cost toyou).
Rubik's Cube BrainTeaser Puzzle wit...
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I'm bored, what do I do now?
Try the Rubik's Revenge, a 4 x 4 x 4 cube. I have written a partially complete solution tothe Rubik's Revenge.
What are some links to other Rubik's Cube pages
These links were all verified December 29, 2006. This is not intended to be a definitive listof all cubing links, rather these have been selected because they are particularlyinteresting, useful, etc.
I. Web Cubes
Michael Schubart's Rubik's Cube Java Applet (highly recommended, veryeasy to use and you can see all six sides at once).Karl Hörnell has a beautiful Java cube (extremely realistic, a bit more difficultto manipulate quickly). I don't think words can quite describe how slick thislooks.This Flash Animation from Eviltron is funny as well as practical. There's alittle song and animation if you solve the cube. I did find the controls a littlefussy, but you probably get used to it after a while. This is definitely worthchecking out.Dan Knights wrote an online Java cube that is now being hosted by
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rubiks.com. It also allows you to solve arbitrary cube configurations (althoughit appears you have to pay for this feature) and demonstrate solutions. Highlyrecommended.Eric Dietz has written a Rubik's Cube Solver. You can enter the entire state ofthe cube and the program will list a sequence of moves to solve the cube, withavailable diagrams for each step. There is also C++ source code available ifyou are interested in programming.
II. General Cube Pages
Georges Helm has an interesting Rubik's cube page with a lot of links andhistorical information.There is now an Official Rubik's Cube Site which is pretty slick, but there isnot very much useful content, IMHO.Also try Matt Monroe's page, which has s instructions for solving the Rubik'sCube, among other things. He also has solutions for the Professor's Cube (5 x5 x 5), Square 1 and Pyramix.Jin "Time Traveler" Kim has a Rubik's FAQ which covers a large number ofRubik's type puzzles. This page hasn't been updated in a long time but there isa lot of information you won't find anywhere else, particularly on otherRubik's Cube-like puzzles.Visit the The Domain of the Cube, an incredibly useful resource with lots ofinfo and links.Martin Schöenert has an archive from the Cube-Lovers mailing list.Chris Loans has a Rubik's Cube page which also describes Solution #1.Check out the Wikipedia entry on the Rubik's Cube. You can even add or editthe article.Here's a no-notation solution from Chase Kimball and Matthew Campbell.Paolo Scalia has a website called Rubik's Illusions that explains both thelayers method and the Petrus method, with animations.
III. Speed Cubing
Visit the Speed Cubing site maintained by Ron van Bruchem. There are somereally incredible records listed on this site. There's also an RSS feed for their
Rubik's Cube news page. Click this icon to subscribe. Chris Hardwick maintains his Rubik's Cube Page on the speedcubing.comsite.The Rubik's Cube World Records Page only contains records achieved atcompetitions.The stiff_hands cube page has a lot of good speed cubing tips as well as a fastsolution.Lars Petrus has written pages called Solving Rubik's Cube for Speed. Thepages detail a highly optimized solution and the moves are demonstrated withJava cubes. Very neat.Jessica Fridrich, the second place finisher at the 2003 Championships, has avery interesting and informative Speed Cubing Page for those of you thatwant to improve your times.There's a Yahoo! Club devoted to speed cubing, Speed Solving Rubik's Cubewhich is quite active.There's a nice online timer for speedcubing as well as a ton of speedcubing
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videos at StrangePuzzle.com.
IV. Rubik's Cube Pages in Other Languages
Danish (Dansk) (Word Document) (translation of this page)French (Français)German (Deutsch)Italian (Italiano)Japanese (日本語) (translation of this page)Portuguese (Português) (translation of this page)Spanish (Español)
V. Mathematical / Technical Cube Pages
Professor W. D. Joyner has several excellent pages, particularly hisPermutation Puzzles Page and his Lecture Notes on the Mathematics of theRubik's Cube.David Miller has a fascinating page titled Solving Rubik's Cube Using the"Bestfast" Search Algorithm and "Profile" Tables. I would imagine this pagewould appeal to those with an interest in AI.Here are some extensive notes for a course called Mathematics of the Rubik'sCube at Stanford.
What are some games I can play with the cube?
An interesting game for two people is to get a correct cube, and for one person to secretlymake a number of moves (try 4 to start with), and then the other person has to undo thosemoves. I've been able to do 7, maybe about 25% of the time. I can do 6 about 75% of thetime, and 5 or less virtually always.
There are numerous patterns that can be made with a correct cube. Perhaps the twosimplest are the "checkerboard" pattern, and the "dots" pattern, which are achieved byapplying the following moves to a solved cube. For the checkerboard, do: R2 L2 U2 D2 F2
B2. For the dots pattern, do: R L' F B' U D' R L'. There are hundreds of other patterns totry -- The Domain of the Cube has a huge collection.
How do I orient the center pieces?
Some promotional or novelty cubes contain patterns on the stickers. Using the methodabove you can solve all of the cube as normal, except you may end up with the centerpieces incorrectly oriented. If this is the case you can use a one or more of these threeprocesses to correctly orient them.
Rotate the U center 180 degrees: U R L U2 R' L' U R L U2 R' L'Rotate the U center clockwise 90 degrees and the F center anti-clockwise 90degrees: F B' L R' U D' F' U' D L' R F' B URotate the U center clockwise 90 degrees and the D center anti-clockwise 90degrees: R L' F2 B2 R L' U R L' F2 B2 R L' D'
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How do I tell if the cube is unsolvable from a given state?
As mentioned above, if you disassemble the cube and reassemble it randomly, it is mostlikely that it will be impossible to solve. Try to solve the cube as normal. All of the firstfew steps should work. When you get down to the final stage for edge pieces or cornerpieces, and you notice one of the following oddities, you will need to disassemble the cubeand reassemble it properly:
One edge piece is flipped in place and all other pieces are correct.Two edge pieces need to be swapped and all other pieces are correct.One corner piece needs rotating and all other pieces are correct.Two corner pieces need to be swapped and all other pieces are correct.
These are simplest cases. For the corner rotating one, if you count a correct piece as 0, apiece that needs rotating clockwise as -1 and a piece that needs rotating counter-clockwiseas -1, then the sum of all corner pieces must be divisible by 3. Similar arguments can beestablished for the other two conditions.
How do I solve a Sudokube?
There appear to be a few Rubik's Cube-like puzzles shamelesslycashing in on the Sudoku craze. I have one such puzzle, which issimply a Rubik's Cube with each side labelled with the numbers 1through 9. To solve the Sudokube, you have to get all numbers inorder on each side, and all right-side-up (not a concern with the colorson a regular Rubik's Cube). There doesn't appear to be much in theway of actual Sudoku-playing in this puzzle, although, as withSudoku, you do have to get one of each number in a 3 x 3 box.
It's actually a bit complicated, and a bit lame, in my humble opinion. Ihave one of these and the cube is of poor quality. The faces do notturn smoothly and there is a defect in the plastic in one of the cornerpieces. Overall it feels kind of cheap, but it is functional. It's a bitbigger and it has lower density than a regular cube. I've included a linkon the right here, however I do not have this exact model (same layoutthough). I bought mine at Chapters for $5 and it says "Distributedunder license from Product Creations Limited, UK although I can'tfind it on their website.
That being said, the object is to put all of the numbers from 1 to 9 inorder on each side of the cube. It's a bit more difficult than solving aregular Rubik's Cube, due to the following three basic observations:
There is an orientation required for each of the center pieces, alllabelled with a 5.There are two groups of ambiguous pieces: four 4/6 edge piecesand two 2/8 edge pieces. You don't know which of these piecesis which.There are two ambiguous pieces: 2/2 and 8/8, for which the
SudoKube, Sudoku, 3DPuzzle
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correct orientation of these pieces is not clear.
Take a quick look at how the sudokube is set up before you scramble it. Note that the toplayer (if you take the 5 with the Sudokube logo as the top) corner pieces all have 3/1 ontwo sides, and in the bottom layer they all have 7/9 on two sides. All the top layer edgepieces have a 2, and all the bottom layer edge pieces have an 8. Strangely, the 9s areunderlined, presumably to distinguish them from the 6s, but it's not necessary because allof the 6s are on edge pieces and all of the 9s are on corner pieces.
I don't know what officially counts as "cheating" but if you are just trying to get it done, Iwould recommend putting a small bit of tape on the U and D sides of the 2/2, 8/8 and both2/8 pieces, if you are starting with a solved sudokube. This way you will know how theseare oriented. Also, you will need a way of distinguishing between the 4/6 pieces. I putsome tape on the 4 and 6 faces one one side of the cube (call it F) in a particular spot, thenput some tape on the 4 and 6 faces on the opposite site (i.e. B) in a different spot. The 4and 6 faces on the L and R sides (given that the U face is the side that has the 5 with theSudokube logo) do not have any tape. If you do this, you can simply solve the cube asnormal, and then just follow the instructions in How do I orient the center pieces? to finishthe cube. You can start with the side that includes the 5 with sudokube text underneath it,to keep track of which side is which.
If you are given a sudokube in a scrambled state, you will have to engage in some trialand error (which is not in the spirit of the original puzzle, in my opinion). Try solving thecube as normal. You may come across a situation that seems impossible, for instance, asituation in which all of the edge pieces are solved, and all of the corners appear to besolved, except for two corners that need swapping locations. If this happens, then you havea couple of edge pieces swapped (although it's not possible to tell because the pieces lookthe same). If you are not using tape, you'll have to immediately swap the 2/8 pieces andtry again, until you either solve the cube or reach some sort of impossible situation, uponwhich you will have to try another combination of some of the ambiguous pieces. If youare using tape, place some tape on of the pieces to remind you which orientation of thosepieces you were using, and try again. This is definitely quite frustrating without tape.
Please let me know if you find a good straightforward solution that does not require tapeor backtracking.
I made some videos of me talking about some of these things and solving the Sudokube.
Here are some other possible misspellings, so people can find this page with a searchengine: suduko cube, sudocube, sudoku kube, su doku, soduko, soduku, sodoku.
MiscellaneousPersonal Records
My personal best time for solving the Rubik's Cube is 36.7 seconds, set on April 21, 2007,using a combination of solution #2 and the Lars Petrus solution. This was a fairly luckycase, I'll admit, as a couple of steps were skipped, but still a legitimate solve. My recordwith Solution #1 is 57 seconds and my record with Solution #2 is 40 seconds. I also keep arecord for solving 5 straight cubes. My record, set on February 13, 1998 is 286 seconds
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(average of 57.2 seconds per cube).
On February 1, 1996 I managed to do the cube in only 6 "looks". I haven't really tried tobreak this record since then, it is too taxing on the mind and I don't have a photographicmemory or anything like that. Here's where I took my "looks": one at the start, then Isolved the edge pieces on the first face; one look to solve the corner pieces of that firstface; two looks for the middle edge pieces; one look for Step 3 and one look for the lastthree steps (note that they Step 4 is totally independent from Steps 5 and 6, and it'spossible to look ahead and see which corner pieces will need to be rotated in whichdirections).
Page History
A short note about the history of this page. The previous locations of this page werehttp://qlink.queensu.ca/~4mj2/rubiks.html (original) andhttp://www.ncf.carleton.ca/~ad161/rubiks.html. I hope that this current URL,http://jeays.net/rubiks.htm will be this page's permanent home. I created the pagein November 1995 during a period of renewed interest in the cube for me. I did have acounter on my home page at that time, which was getting around 5 hits per day. I assumedthat the traffic on the cube page would be a subset of that. I was extremely surprised whenI put a hit counter on the page in October 1997 to see that it was getting around 50 hits perday. This page underwent a major overhaul in December 1997 when I added the secondsolution. By the time that my university account expired in April 1999, the page wasreceiving around 120 hits per day.
By August 2002, this page, and its previous revisions had received somewhere on theorder of 250,000 hits, about 120,000 of them on http://jeays.net, which I started inJanuary 2001. In 2006 the page received approximately 750,000 views. Frankly, ifsomeone told me that the page would get 1000 hits the night, in late 1995, after I wrotethe initial version (probably while procrastinating over doing my homework), I don't thinkI would have believed them. Thanks for coming, and I hope you'll continue to use the pageand recommend it to your friends that are interested in the cube.
Poll Archives
February 2005
I think the poll software had some serious glitches for this one. Some of the vote totalswere going down from day to day. In any case, the question was 'How old were you whenyou first solved the cube?' and approximately 2/3 of respondents said 0-17, 1/5 said 18-29and the remaining responses were distributed fairly evenly among the other age groups,30-44, 45-64 and 65+. The final totals on the poll said over 90% were in the 0-17 categorybut I'm almost certain that this was due to a software flaw. I may re-run this poll onemonth when I get some better poll software. In any case, I can't say I'm overly surprised atthe results since the cube came out in the early 80s, which means that everybody youngerthan about 40 probably had access to the cube in the 0-17 age range.
March 2005 - November 2005
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I know, I was supposed to do a poll every month. I guess that didn't quite happen. Thequestion was: ' How many people have you taught to do the cube?'. The final results forthis poll over 16,000 votes were as follows: 60% said nobody, 11% said one person, 8%said 2-4 people, 6% said 5-9 people and 13% said 10 or more. I'm a little skeptical aboutthe number of people teaching 10 or more, but it's always possible.
December 2005 - December 2006
There was a poll about how many cubes you have owned. I think the polling software isincredibly inaccurate, so this is the last poll for now. The results were probably somethinglike this: 20% have never had a cube, 50% have had one, 35% have had 2-4, and theremaining 10% have had 5 or more.
Web Tools That Reference This Page
Alexa: Overview, Write a Review, Traffic DetailDigg.com: diggs (bookmarks to this page)Google: pages that link here, similar pagesWayback Machine: Older versions of this page
Who's amung us: Number of current viewers
Contact Me
Comments, Questions, Feedback, Concerns, Musings, Observations, New Moves, RecordTimes You'd Like to Share With Someone Who Might Care, Criticisms, Observations ofErrors in the Text and Flames can all be directed to rubiks [at] jeays [dot] net,and I'll be happy to answer these. I do get some emails with generic complaints about theinstructions, but I can't help you if you don't have a specific question. Tell me what youtried and what went wrong, and then I should be able to help you.
Try checking out my How to Solve the Rubik's Cube Playlist on YouTube if you want tosee the moves for Solution #2 in action.
If you are down to the last few pieces and your cube is in a position for which there doesnot appear to be an appropriate move, you might want to check the list of unsolvablepositions. Please note that I do not have any solutions to any other puzzles (other than thepartial solution to the Rubik's Revenge).
N'hésitez pas à communiquer avec moi en français.
One more thing, I guess it's too bad that there isn't a "URL" piece...oh, forget it!
Copyright © 1995-2007 Mark Jeays. Permission is granted to copy, distribute and/ormodify this document under the terms of the GNU Free Documentation License, Version1.1 or any later version published by the Free Software Foundation.
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